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A Note on Endomorphisms of Local Cohomology Modules (1405.1249v2)
Published 6 May 2014 in math.AC
Abstract: Let $I$ denote an ideal of a local ring $(R,\mathfrak{m})$ of dimension $n$. Let $M$ denote a finitely generated $R$-module. We study the endomorphism ring of the local cohomology module $Hc_I(M), c = \grade (I,M)$. In particular there is a natural homomorphism $\Hom_{\hat{R}I}(\hat{M}I, \hat{M}I)\to \Hom_{R}(Hc_{I}(M),Hc_{I}(M))$, where $\hat{\cdot}I$ denotes the $I$-adic completion functor. We prove sufficient conditions such that it becomes an isomorphism. Moreover, we study a homomorphism of two such endomorphism rings of local cohomology modules for two ideals $J \subset I$ with the property $\grade(I,M) = \grade(J,M)$. Our results extends constructions known in the case of $M = R$ (see e.g. \cite{h1}, \cite{p7}, \cite{p1}).